Let us assume that \(\mathfrak {R}\)  is not known analytically and we cannot numerically generate samples from it directly. Thus we have to estimate the distribution from the data. Algorithm REF  summarizes the basic computations for computing a reduced order model. It returns the deterministic part of the model, the residual noise sequence, the modal noise sequence, and the innovation noise sequences. We note that while we are working with i.i.d. stochastic dynamical systems, and therefore should be working with the stochastic Koopman operator [1]}, (REF ) computes the reconstruction coefficients along a single realization of evolution of the dynamical system. Wanner and Mezić [2]} show the convergence of these coefficients to the true reconstruction coefficients of the stochastic Koopman operator using an ergodicity argument in the large data limit \(T\rightarrow \infty \) .
